Line Integrals: Compute 4xz + 2y dx

In summary, the conversation is about a person needing help with a problem involving computing a line integral. They are given a line segment and must use a specific formula to solve the problem. They discuss parametricizing the line and dealing with a "ds" term.
  • #1
PhysicsMajor
15
0
Greetings All,

I need serious help with this problem , well maybe several problems but I'll take it one at a time:

Compute the line integral of (4xz + 2y)dx where C is the line segment from (2,1,0) to (4,0,2).

Thanks
 
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  • #2
[tex] \int_c f(x,y,z) ds = \int_a^b f(x(t), y(t), z(t)) \sqrt { \left( \frac{dx}{dt} \right)^2 + \left( \frac{dy}{dt} \right)^2 + \left( \frac{dz}{dt} \right)^2 } [/tex]

so, you need to parametricize your line in space. par exemple:

[tex] \begin{align*}
x=2 + 2t \\
y=1 - t \\
z=2t \\
0<t<1
\end{align*}
[/tex]

those are *supposed* to be "less than or equal to" signs, but i never was able to find those...

anyhow, you ought to be set to solve this puppy, unless you made it to here and collapsed...
 
Last edited:
  • #3
Well, he didn't have any annoying [itex]ds[/itex] to deal with.

All he needs to do is parameterize the curve and plug in for x, y, z, dx.
 
  • #4
oh, indeed, it is a dx. ecks, ess, they sound so similar in my noisy head...
 
  • #5
What a strange problem.
 

What is a line integral?

A line integral is a type of integral that calculates the total value of a function along a given curve or path. It takes into account both the function and the path, and is often used in physics and engineering to calculate work, flux, and other physical quantities.

What is the formula for computing a line integral?

The formula for computing a line integral is ∫C f(x,y) ds, where C is the curve or path, f(x,y) is the function being integrated, and ds is an infinitesimal element of length along the curve.

How do you compute a line integral numerically?

To compute a line integral numerically, you can divide the curve into smaller segments and approximate the integral using the trapezoidal rule or Simpson's rule. Alternatively, you can use software programs such as Mathematica or MATLAB to calculate the integral.

What is the significance of the function 4xz + 2y in the given line integral?

The function 4xz + 2y represents the value of the function along the given curve or path. In this case, it is the function that is being integrated, and its value will change depending on the values of x and y at each point along the curve.

What are some real-world applications of line integrals?

Line integrals have many real-world applications, such as calculating the work done by a force along a curved path, finding the flux of a vector field through a closed surface, and determining the mass or charge of an object by integrating its density or charge distribution along a path.

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